Comptes Rendus
no. 8
Combinatorics/Number Theory
On multiple sum and product sets of finite sets of integers
[Sur les ensembles de sommes et produits multiples d'ensembles finis d'entiers]

Présenté par : Jean Bourgain

Jean Bourgain1 ; Mei-Chu Chang2
1 Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA
2 Mathematics Department, University of California, Riverside, CA 92521, USA
Comptes Rendus. Mathématique, Volume 337 (2003) no. 8, pp. 499-503.

Soit A⊂ℤ un ensemble fini d'entiers et |A|=N⩾2. Pour tout entier positif k, denotons kA (resp. A(k)) l'ensemble de toutes les sommes (resp. produits) de k éléments de A. On démontre que pour tout b>1, il existe k=k(b) tel que max(|kA|,|A(k)|)>Nb. Ceci répond affirmativement à des questions posées dans Erdős et Szemerédi (Stud. Pure Math., 1983, pp. 213–218), Elekes et al. (J. Number Theory 83 (2) (2002) 194–201) et, récemment, par S. Konjagin (communication privée). La méthode est basée sur des arguments d'analyse harmonique dans l'esprit de Chang (Ann. Math. 157 (2003) 939–957) et de la combinatoire sur des graphes.

Let A⊂ℤ be a finite set of integers of cardinality |A|=N⩾2. Given a positive integer k, denote kA (resp. A(k)) the set of all sums (resp. products) of k elements of A. We prove that for all b>1, there exists k=k(b) such that max(|kA|,|A(k)|)>Nb. This answers affirmably questions raised in Erdős and Szemerédi (Stud. Pure Math., 1983, pp. 213–218), Elekes et al. (J. Number Theory 83 (2) (2002) 194–201) and recently, by S. Konjagin (private communication). The method is based on harmonic analysis techniques in the spirit of Chang (Ann. Math. 157 (2003) 939–957) and combinatorics on graphs.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2003.08.010
Jean Bourgain 1 ; Mei-Chu Chang 2

1 Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA
2 Mathematics Department, University of California, Riverside, CA 92521, USA 
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Jean Bourgain; Mei-Chu Chang. On multiple sum and product sets of finite sets of integers. Comptes Rendus. Mathématique, Volume 337 (2003) no. 8, pp. 499-503. doi : 10.1016/j.crma.2003.08.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.08.010/

[1] J. Bourgain On the Erdős–Volkmann and Katz–Tao Ring Conjectures, Geom. Funct. Anal., Volume 13 (2003)

[2] M. Chang The Erdős–Szemerédi problem on sum set and product set, Ann. of Math., Volume 157 (2003), pp. 939-957

[3] G. Elekes; M. Nathanson; I. Ruzsa Convexity and sumsets, J. Number Theory, Volume 83 (2000) no. 2, pp. 194-201

[4] P. Erdős; E. Szemerédi On Sums and Products of Integers, Stud. Pure Math., Birkhäuser, Basel, 1983 (pp. 213–218)

[5] S. Konjagin, Private communication

[6] M.B. Nathanson Additive Number Theory, Inverse Problems and the Geometry of Sumsets, Graduate Text in Math., 165, Springer-Verlag, New York, 1996

[7] W. Rudin Trigonometric series with gaps, J. Math. Mech., Volume 9 (1960), pp. 203-227

[8] J. Solymosi, On the number of sums and products, Preprint, 2003

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